Thankfully, recall our previous informal discussion on stronger arguments and weaker arguments in Section , there exactly exists a quite handy and natural notion of partial order among arguments, namely by order of argumentation strength. And then immediately, our core philosophical idea bears intimate similarity to the ultrafilter extension construction in modal logic, so that proper filters correspond to “consistent theories” while ultrafilters correspond to “maximal consistent theories”, for readers unfamiliar with this topic, cf. [1]} as a rudimentary introduction. Formal definitions go as the following:
[Preorder \(\le \) ]
Given an argument Kripke model \(\mathfrak {N}=(\mathbb {W},A,G)\) , we define a preorder \(\le \subseteq \mathbb {W}\times \mathbb {W}\)  such that for any \(U,V\in \mathbb {W}\) , \(U\le V\)  iff \(\forall p\in P\forall W\in \mathbb {W}\) , \((W,V)\in A(p)\rightarrow (W,U)\in A(p)\) .

Verifying that \(\le \)  is reflexive and transitive is obvious. As for the intuitive reading, \(U\le V\)  means “argument \(U\)  is no weaker than argument \(V\)  in attack power”, just as Definition  shows, if and only if any pair of proposition and argument attacked by \(V\)  also gets attacked by \(U\) .

\(U\le V\)  intuitively means “\(U\)  is stronger than \(V\) ” (or at least equally strong), not “\(V\)  is stronger than \(U\) ”, and therefore, here it is inadvisable to read the notation “\(\le \) ” as “less than or equal to”. This kind of definition in many ways conforms to the convention in set theory [2]}: the “smaller”, the stronger.

Given any fixed partial order (or more generally, fixed preorder) \(\le \)  over any fixed nonempty set \(\mathbb {W}\) , the usual definitions of filters and ultrafilters [3]} are presented as below:
[Filter]
For any \(\mathbb {U}\subseteq \mathbb {W}\) , \(\mathbb {U}\)  is a filter iff:
